I would like to understand the asymptotic behaviour of the Fourier coefficients of
power type functions
$f(t) = |t|^{-\alpha} 1_{[-\pi, \pi]} \qquad 0 < \alpha<1.$
I suppose this is a classic result that I am supposed to know which can be found in many books, but I do not know where to start reading. Can you give me a hint please?

Just did some crude numerics on that for fun, result is that $\hat f(n) \approx n^{-(1 - alpha)}$. Code (python/numpy) is at pastebin.com/rL5QNMnv if you want to take a look. I'm interested in the proof.
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Antoine LevittDec 14 '11 at 15:50

Actually, this asymptotic behaviour is quite easy to prove using a simple change of variable. It doesn't give the full asymptotics and constants though, see Igor Rivin's answer below for that.
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Antoine LevittDec 14 '11 at 16:02

@Antoine I am amused that pastein.com is just like mathurl...
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Igor RivinDec 14 '11 at 16:04

There's a lot of sites like that, usually used for pasting code. I didn't actually know about mathurl, it's pretty neat!
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Antoine LevittDec 14 '11 at 16:08

This is a cosine transform of $t^{-a} \mathbb{1}[0, 1],$ which can be evaluated explicitly using the trusty mathematica, which gives:
$
\frac{t^{a+1} \,
_1F_2\left(\frac{a}{2}+\frac{1}{2};\frac{1}{2},\frac{a}{2}+\frac{3}{2};-\frac{1}{4}
k^2 t^2\right)}{a+1}
$
If you want the asymptotic of the above expression in $k$ (the transform variable), you can use the mathematica command Series[your_favorite_expression, {k, Infinity, 10}] (10 gives you the first ten terms in the power series, feel free to use your favorite integer). If you use 1 instead of 10 (for ease of typesetting), you get this.
(sorry, easier to use mathurl than do line breaks by hand).